Vaughan wrote:
What's odd is that the term seems to be used precisely when it is mathematically inappropriate in the above sense (quite apart from medical or sensitivity issues).
The real manifestation of the "same" entity is not with objects at all but with homsets, namely the homset C(D'(d'), d) in C and the homset C'(D(d), d') in C', which *do* have the same number of morphisms.
If anything deserves the epithet in question it is that homset in each category. The two homsets are in bijection, but their targets don't correspond, having only in common that they are the dualizers in the respective categories.
Yes, I always thought it odd that even when one wants to accept category-theoretic foundations (e.g. ETCS or similar), then suddenly something like this comes along, where people start saying there is a thing which is an object of two different categories. Such a property isn't even expressible in type theory-style foundations, where even elements of two different sets aren't comparable... But since both of the categories in each pair Vaughan mentioned are enriched over Set, we *are* allowed to compare hom-sets, at least using some sort of roughly canonical isomorphism. Using the formalism suggested (hom-sets), it seems much easier to set down a definition of these slippery objects. And (more whimsically) regarding terminology: if someone wanted to use the analogy of a door, why not a window? We can have fenestral objects, by which one can 'see' from one category to another. And unfortunately, 'liminal' gives rise to subliminal, which might be a natural prefix extension mathematically but is even more confusing than the existing inaccurate term. :-) David [For admin and other information see: http://www.mta.ca/~cat-dist/ ]